7 Mathematical skills

Exam analysis

Most tested topics

1 Arithmetic and numerical computation

A decimal records place value on both sides of the decimal point. Keep digits aligned by place value during addition or subtraction, and preserve units before carrying out a biological calculation.

Move	Example
identify place value	in 0.025, 2 is hundredths and 5 is thousandths
align or convert units	90 cm = 0.90 m before combining it with a speed in m s⁻¹
calculate with the full values	0.25 m ÷ 0.005 s = 50 m s⁻¹
check position of the decimal point	dividing by a number smaller than 1 should increase the numerical value
report meaning and unit	write 50 m s⁻¹, not an unlabelled 50

A zero before the decimal point makes values below one unambiguous: write 0.5, not .5. Trailing zeros may communicate recorded precision—9.0 mg is expressed more precisely than 9 mg—even though the numerical value is equal.

Do not align numbers by their final digit or remove zeros before deciding what they communicate. A correct decimal with mismatched units is still a wrong biological calculation.

Standard form writes a non-zero number as $a \times 10^n$ , where $1 \leq |a| < 10$ and $n$ is an integer. The coefficient shows the significant digits; the power of ten shows scale.

Operation	Rule	Example
decimal to standard form	move the point until the coefficient is between 1 and 10; count places	$480000000 = 4.8 \times 10^8$
standard to decimal	positive power moves right; negative power moves left	$2.0 \times 10^{-1} = 0.20$
multiply	multiply coefficients and add powers	$(5.0 \times 10^9)(5.0 \times 10^3)=25 \times 10^{12}=2.5 \times 10^{13}$
divide	divide coefficients and subtract powers	$(6.0 \times 10^8)/(3.0 \times 10^2)=2.0 \times 10^6$
add or subtract	first express terms with the same power of ten	$3.2 \times 10^6+0.5 \times 10^6=3.7 \times 10^6$

After calculating, renormalise the coefficient. $25 \times 10^{12}$ has the right value but is not standard form because 25 is not between 1 and 10.

Ratios, fractions and percentages express one quantity relative to another. The denominator or comparison base must match the biological question before any arithmetic is performed.

Need	Relationship	Worked biological example
simplify a ratio	divide every part by the same factor	21 yellow : 10 brown = 2.1 : 1
find a fraction of a total	fraction × total	half of 65 million = 0.5 × 65 million = 32.5 million
find a percentage of an amount	percentage ÷ 100 × amount	16.7% of 28.0 g = 0.167 × 28.0 = 4.68 g
express a part as a percentage	part ÷ whole × 100	56 farmed out of 146 total = 38.4%
percentage change	change ÷ original × 100	from 0.70 to 10.0 million: 9.30 ÷ 0.70 × 100 = 1330% approximately
powers and roots	a power repeats multiplication; a root reverses that power	$3^2=9$ and $\sqrt{9}=3$

For percentage change, divide by the original value, not the final value. A percentage may exceed 100% when the increase is greater than the original. Keep ratios in the requested order and attach units to calculated amounts.

An estimate replaces values with nearby easy numbers to find an approximate result without a calculator. Its purpose is to predict the scale and expose implausible calculator entries, not to replace a required accurate answer.

Move	Example using $(4.8 \times 10^8)(0.0040)$
round to one useful significant figure	$4.8 \times 10^8 \approx 5 \times 10^8$ and $0.0040 = 4 \times 10^{-3}$
calculate mentally	$(5 \times 4) \times 10^{8-3}=20 \times 10^5$
write the scale clearly	$20 \times 10^5=2 \times 10^6$
compare with the accurate result	$1.92 \times 10^6$ is close to $2 \times 10^6$ , so its order of magnitude is plausible

Round enough to make the arithmetic simple but keep the important scale. If one value is just above and another just below a convenient number, note whether both rounding choices push the estimate in the same direction.

An estimate should not contain more apparent precision than the original calculation. A result of $2.000 \times 10^6$ is not a sensible estimate here; write about $2 \times 10^6$ and retain the approximation sign or wording.

2 Handling data

Significant figures count meaningful digits from the first non-zero digit. A final biological result should not imply finer precision than the measurements used to calculate it.

Value	Significant figures and reason
0.00450	3; leading zeros locate the decimal point, while the final zero is significant
1200	ambiguous without context; $1.2 \times 10^3$ states 2 significant figures
12.46 to 3 s.f.	12.5 because the next digit is 6, so 4 rounds up
0.06784 to 2 s.f.	0.068 because counting starts at 6 and the next digit rounds 7 up

Keep extra digits during intermediate steps and round once at the end. Match the requested significant figures and retain the unit.

Decimal places count positions after the decimal point; significant figures count meaningful digits. For example, 0.0125 has four decimal places but three significant figures.

The arithmetic mean shares the total of all included observations equally across their number. It summarises the centre of repeated numerical data but does not show their spread.

$mean = sum of included values / number of included values$

For gelatine volumes 0.55, 0.54 and 0.61 cm³, the mean is $(0.55+0.54+0.61)/3=0.5666…$ cm³, reported as 0.57 cm³ to two decimal places.

Count only the values actually included. Do not discard an anomalous result merely because it changes the mean; exclusion needs a recorded evidence-based reason, and the reported mean should make that decision clear.

A bar chart compares numerical values for distinct categories. Because the categories are separate rather than continuous, the bars are separated by gaps.

Construct	Interpret
place categories on the horizontal axis	identify the named group represented by each bar
label the vertical axis with variable and unit	read values from the scale, not from apparent bar area
start at zero unless a clearly shown break is justified	compare heights using the same scale
use equal bar widths and gaps	state the largest, smallest or difference with values

If unlikely and likely groups have LH concentrations of 5 and 45 arbitrary units, the likely category is 40 units higher and nine times the unlikely value.

Do not join category bars into a continuous shape. A histogram looks similar but represents continuous grouped intervals, so its bars touch and may use frequency density.

A frequency table records how many observations fall in each value or class interval. A histogram displays grouped continuous data with touching bars whose areas represent frequencies.

Data move	Rule
define classes	make intervals non-overlapping and cover every possible value
tally observations	place each observation in exactly one class
record frequency	count the tally in each class and check the total equals the sample size
draw equal-width histogram classes	bar height may be frequency because equal widths preserve area comparisons
draw unequal-width classes	use frequency density = frequency ÷ class width, so bar area equals frequency

Bar height alone does not represent frequency when class widths differ. Histograms are for continuous grouped measurements; ordinary bar charts are for separate categories.

Sampling measures a manageable subset to infer properties of a larger population. The sample must be selected without systematic bias and be large and repeated enough to capture natural variation.

Situation	Defensible sampling design
organisms across an area	overlay a grid, choose coordinates randomly, use equal-sized quadrats and count consistently
change along a gradient	place quadrats at fixed intervals along a transect
mobile organisms	use a defined trapping method for equal times and avoid counting the same individual twice where possible
microscopic density	count in known equal areas, repeat across randomly selected fields of view, find mean density and scale to total area

Two stomata in $0.4 \text{ mm} \times 0.4 \text{ mm}=0.0016$ cm² give a density of $2/0.0016=1250$ stomata cm⁻². Applied to 150 cm², the estimate is 187500 stomata.

A large convenient sample can still be biased. More repeats improve representation only when locations, times or individuals are selected by a method appropriate to the population and question.

Probability measures how likely an outcome is, from 0 for impossible to 1 for certain. For equally likely outcomes, it is the number of favourable outcomes divided by the total number of possible outcomes.

$P(event) = favourable outcomes / total possible outcomes$

For a heterozygous monohybrid cross $Aa \times Aa$ , one of four equally likely genotype outcomes is $aa$ , so $P(aa)=1/4=0.25=25\%$ . For two independent events, multiply their probabilities.

A probability predicts a long-run proportion, not an exact result in a small family or sample. Observed frequencies may differ by chance, and events must be independent before their probabilities are multiplied.

The median is the middle value after numerical data are ordered; the mode is the value or category that occurs most often. They answer different questions about a dataset.

Measure	Method	Useful when
median	order values; choose the middle, or average the two middle values when the count is even	extreme values would pull the mean away from a typical central value
mode	count occurrences and select the most frequent value or category	the most common outcome matters, including non-numerical categories

For 6, 8, 8, 9, 16, the median is 8 and the mode is 8. For 6, 8, 9, 16, the median is $(8+9)/2=8.5$ and there is no mode because no value repeats.

The median cannot be found from unordered positions, and a dataset may have no mode or more than one mode. Do not call the largest value the mode unless it is also most frequent.

A scatter diagram plots paired measurements for two numerical variables. The overall point pattern can show positive correlation, negative correlation or no clear correlation.

Pattern	Interpretation
points rise from left to right	positive correlation: larger values of one variable tend to accompany larger values of the other
points fall from left to right	negative correlation
points show no direction	no clear correlation in the observed range
points lie close to a trend	stronger correlation than a widely scattered pattern
one point lies far from the pattern	possible anomaly; check it without deleting it automatically

Plot the independent variable on the horizontal axis and its paired dependent value vertically. A best-fit line or curve follows the overall pattern rather than joining each point.

Correlation alone does not prove that one variable causes the other. A third variable, reverse causation or chance may explain the pattern, and extrapolation beyond the observed range is uncertain.

An order of magnitude is a factor of ten. Expressing a quantity in standard form reveals its scale and allows rapid comparisons between very large or very small biological values.

Move	Example
express each value in standard form	a bacterium $2 \times 10^{-6}$ m; a cell $2 \times 10^{-5}$ m
compare powers of ten	exponents differ by $-5-(-6)=1$
convert exponent difference to a factor	$10^1=10$ , so the cell is about ten times longer
estimate a product or quotient	combine rounded coefficients and add or subtract exponents

A difference of two orders of magnitude means a factor of $10^2=100$ , not a difference of 2. Coefficients near a power boundary can affect the nearest order, so keep them visible until the comparison is justified.

3 Algebra

Changing the subject rewrites an equation so the required symbol stands alone while the relationship remains equivalent. Apply inverse operations to both sides, preserving brackets and powers.

Starting relationship	Required subject	Equivalent form
$c=a/v$	$v$	multiply by $v$ , then divide by $c$ : $v=a/c$
$r=d/t$	$t$	$t=d/r$
$B=m/h^2$	$m$	$m=Bh^2$
$B=m/h^2$	$h$	$h=\sqrt{m/B}$ for a positive physical height

Urine concentration is $c=a/v$ . With amount $a=600$ milliosmoles and maximum concentration $c=1400$ milliosmoles dm⁻³, first rearrange to $v=a/c$ , then calculate $v=600/1400=0.429$ dm³, or about 429 cm³.

Moving a term across an equals sign is shorthand for applying an inverse operation to both sides. Do not change a sign or invert a quantity without showing the operation that keeps the equation balanced.

Substitution replaces each symbol with its numerical value. Before calculating, convert quantities to the units required by the equation and use brackets so powers and denominators apply to the intended value.

$BMI = mass in kg / (height in m)^2$

Move	BMI example
identify symbols and required units	mass = 60 kg; height must be in metres
convert units	165 cm = 1.65 m
substitute with brackets	$BMI=60/(1.65)^2$
calculate in operation order	$(1.65)^2=2.7225$ , then $60/2.7225=22.0$
interpret only after calculation	22.0 lies in the stated healthy category

Do not square only part of a substituted height or mix centimetres with a formula defined in metres. Units are part of the input, not decoration added after arithmetic.

Solving an equation finds the numerical value of an unknown that makes both sides equal. Undo operations in reverse order and perform the same operation on both sides.

Equation	Balanced steps	Solution check
$3x+6=24$	subtract 6: $3x=18$ ; divide by 3: $x=6$	$3(6)+6=24$
$y/5-2=4$	add 2: $y/5=6$ ; multiply by 5: $y=30$	$30/5-2=4$
$2z^2=50$	divide by 2: $z^2=25$ ; take the relevant root	for a positive biological length, $z=5$

When an equation contains measured quantities, retain units through the solution and judge whether negative or alternative roots make physical sense in that context.

Changing the subject produces a formula in symbols; solving produces a value after known quantities are supplied. Always substitute the result back into the original equation to detect an arithmetic or sign error.

4 Graphs

Translating between graphical and numerical form preserves the variables, units and scale while changing how the relationship is represented. Read coordinates from the axes before describing a pattern.

Graph-to-number move	Number-to-graph move
identify the horizontal variable, vertical variable and units	put each paired observation into an $(x,y)$ row
locate a stated $x$ value and move vertically to the plotted point or line	choose an axis range and regular scale covering all values
move horizontally to read the corresponding $y$ value	plot each pair at its coordinate
calculate a difference or rate from selected coordinates when required	use a key when several datasets share the axes
describe direction, range, plateau or fluctuation with values	retain the exact table values even when the graph summarises their pattern

If a myelinated neurone graph gives 4.4 m s⁻¹ at a diameter of 1.0 µm, the numerical translation is the coordinate $(1.0\,\mu\mathrm{m},4.4\,\mathrm{m\,s^{-1}})$ . State both values and units.

Reading between plotted values is interpolation; reading beyond the measured range is extrapolation and is less secure. Do not infer a value from visual height without checking the scale increments.

A two-variable graph places the independent variable on the horizontal axis and the dependent variable on the vertical axis so every paired observation has one unambiguous coordinate.

Plotting decision	Required feature
choose axes	independent variable on $x$ ; dependent variable on $y$
label completely	variable name and unit on each axis
set scales	linear, regular increments that use much of the available grid and include all values
plot points	small, accurate marks at every coordinate
represent connection	join ordered continuous measurements only when instructed or scientifically justified; otherwise use a suitable best-fit line or leave points unjoined
compare datasets	use distinguishable lines or symbols and a complete key; do not extrapolate beyond the data

For seedling dry mass measured on days 4, 8, 12, 16 and 20, plot day on $x$ and dry mass in g on $y$ . Plot fertiliser and no-fertiliser values on the same scales and label both series so their changes can be compared directly.

Discrete categories do not become continuous merely because points can be joined. A line between time points shows an ordered change; a bar chart is usually clearer for unrelated categories.

For a linear graph, slope measures the change in $y$ per unit change in $x$ , and the vertical intercept is the value of $y$ where the line crosses the $y$ -axis at $x=0$ .

$slope m = change in y / change in x = (y₂-y₁)/(x₂-x₁); linear form y = mx + c$

Move	Example for $y=0.8x+0.2$
select two well-separated points on the straight line	$(1,1.0)$ and $(6,5.0)$
calculate vertical and horizontal changes	$\Delta y=4.0$ and $\Delta x=5$
divide and attach compound units	$m=4.0/5=0.8$ units of $y$ per unit of $x$
read or calculate the intercept	at $x=0$ , $y=c=0.2$
interpret	each one-unit rise in $x$ is associated with a 0.8-unit rise in $y$

Use points on the linear line, not necessarily two noisy raw points. The intercept may have no biological meaning when $x=0$ is impossible or lies outside the measured range; report it mathematically without inventing a biological claim.

5 Geometry and trigonometry

Length is one-dimensional, area measures a two-dimensional surface, and volume measures three-dimensional space. Convert all lengths to the same unit before applying a formula, then square or cube that unit in the result.

Shape or quantity	Formula	Example
rectangle area	$A=lw$	$8\text{ cm}\times3\text{ cm}=24\text{ cm}^2$
triangle area	$A=\tfrac12 bh$ using perpendicular height	$\tfrac12(6\text{ cm})(4\text{ cm})=12\text{ cm}^2$
cube surface area	$S=6s^2$ because a cube has six square faces	for $s=4\,\mu\mathrm{m}$ , $S=6(4^2)=96\,\mu\mathrm{m}^2$
cube volume	$V=s^3$	for $s=4\,\mu\mathrm{m}$ , $V=4^3=64\,\mu\mathrm{m}^3$

Label the required dimension, select the matching formula, substitute one consistent length unit, calculate, and check that the final unit is squared for area or cubed for volume. Surface-area-to-volume comparisons must use compatible units before division.

The sloping side of a triangle is not its height unless it is perpendicular to the chosen base. For a cube, $s^2$ is the area of one face; total surface area is $6s^2$ , while volume is $s^3$ .

7 Mathematical skills

Exam analysis

Most tested topics

1 Arithmetic and numerical computation

Use decimal numbers without losing place value

Write and calculate with standard form

Choose the right proportional calculation

Estimate first to check a calculation

2 Handling data

Report only justified significant figures

Calculate and interpret an arithmetic mean

Use a bar chart for separate categories

Organise grouped data with frequencies and histograms

Take a sample that represents the population

Interpret simple probability as a proportion of outcomes

Use median and mode to describe the centre

Read a trend from a scatter diagram

Compare biological quantities by order of magnitude

3 Algebra

Rearrange an equation before using it

Substitute values only after aligning units

Solve a simple equation by undoing operations

4 Graphs

Translate between a graph and its numerical evidence

Plot two variables on a graph that preserves the data

Find and interpret slope and intercept

5 Geometry and trigonometry

Calculate area, surface area and volume with correct dimensions